How is the kernel of a group action defined?

Question

Question:

Show that the kernel of the group action of $G$ acting on set $A$ is equal to the kernel of the corresponding permutation representation of this action.

I'm lost in this definition as I am only familiar with the definition of a kernel of a homomorphism as $\{g \in G~|~\varphi(g) = I_A\}$ (the elements of the domain in which the image is the identity of the target group)

How is this definition applicable to a map $f: G\times A \to A$ in which the target domain is a set?

Answer 1

Let $G$ be a group acting on $A$.

The kernel of the action is the set $K =\{g \in G; g \cdot a = a , \forall a \in A\}$. Now the corresponding permutation representation is a group homomorphism $\psi : G \to S_A$ given by $\psi (g)(a) = g \cdot a $.

• $K \subseteq \ker \psi$

Let $k \in K$, then for all $a \in A$ we have that $\psi (k) (a) = k \cdot a = a$ thus, $\psi (k) = id_A$ and then $k \in \ker \psi$.

• $\ker \psi \subseteq K$

Let $k \in \ker \psi$ be given. Then for all $a \in A$ we have $$ k \cdot a = \psi (k) (a) = id_A(a) = a $$

thus $k \in K$.

Edit: The kernel of $\psi$ is given by $\ker \psi = \{g \in G ; \psi (g) =id_A\}$.

Answer 2

The kernel of a group action is defined as the set of all group elements which act as the identity. The problem is asking you to show that this definition is related to the kernel of a homomorphism by showing that the kernel of the group action is isomorphic to the kernel of the homomorphism from $G$ onto the permutation group of the set.

Please correct me on anything that might be wrong or ambiguous, since I haven't done this in a while.